PALM SPRINGS — President Reagan and Soviet leader Mikhail S. Gorbachev, in a demonstration of the tentative steps toward improved U.S.-Soviet relations that each has taken in recent months, exchanged New Year's greetings Friday to the Soviet and American people.

Reagan used the Soviet airwaves to gently prod the Kremlin on such still-sensitive issues as human rights and regional conflicts, while expressing U.S. determination to reach agreement on long-range nuclear weapons reduction--perhaps in time for a spring summit in Moscow.

The messages were notable not so much for their content as for the fact that they were broadcast at all--and for what was not said. Missing from each was any of the harsh language that has characterized remarks by Reagan and Gorbachev on the more sensitive issues in the U.S. and Soviet relationship.

And, in a reflection of the improvement that has accompanied the signing last month of the treaty eliminating U.S. and Soviet land-based medium-range nuclear weapons, the Soviet willingness to telecast Reagan's remarks was in sharp contrast to the reception a similar New Year's message was given a year ago.

Then, Gorbachev declared that superpower relations were too poor after the unsuccessful summit in Reykjavik, Iceland, to justify an exchange of New Year's greetings. Accordingly, Reagan tried to have his message delivered by radio over Voice of America, but Soviet officials tried to blot out the broadcast, electronically jamming it in many areas of the Soviet bloc.

This year, Soviet citizens turning on their television sets Friday evening could see this: The chief executive of the United States, sitting before a fireplace and the Nobel Peace Prize that was awarded to Theodore Roosevelt, and addressing them from the Roosevelt Room of the White House.

"Good evening. This is Ronald Reagan, President of the United States. I'm speaking to you, the peoples of the Soviet Union, on the occasion of the new year," he said, commencing a condensed introduction to American society that gave his Soviet audience a glimpse of American holiday traditions and a miniature civics lesson.

"At this time of year, Americans travel across the country--in their cars, by train, or by airplane--to be together with their families," he said. "Most of us celebrate Christmas or Hanukkah. And as part of those celebrations, we go to church or synagogue, then gather around the family dinner table," for a meal "of goose, turkey, or roast beef . . . . "

Holiday Gatherings

Reagan's references to the holiday gatherings in both countries was bound to strike a chord with most Soviet viewers, since the New Year's celebrations are warm family affairs there.

In an implied contrast with the Soviet system, Reagan said that a new American president will be chosen this year.

"I'll still be President next January, but soon after that, the man or woman leading our country will be the one the American people pick this coming November," he said.

Reagan kept his criticism of the Soviet Union to a minimum. He said Americans were concerned "about senseless conflicts in a number of regions."

Without mentioning Nicaragua or Afghanistan, where Soviet-supported governments are facing rebellions aided by the United States, and without any direct reference to the Soviet assistance, the President said:

"In some instances, regimes backed by foreign military power are oppressing their own peoples, giving rise to popular resistance and the spread of fighting beyond their borders. Too many mothers, including Soviet mothers, have wept over the graves of their fallen sons."

In addition, he said, "as you know, we Americans are concerned about human rights, including freedoms of speech, press, worship, and travel . . . We will always speak out on behalf of human dignity."

That emphasis on human rights also may have touched a sensitive point so far as the Kremlin is concerned, but it was too general to arouse any reaction among the public, Western observers here said.

The President also described the space-based missile defense program--the Strategic Defense Initiative--as "a defensive shield that will threaten no one." The program is strenuously opposed by the Soviet Union.

Praises Pact

Gorbachev, noting the improved ties that have stemmed from the medium-range arms treaty, said the pact "marks the first step along the path of reducing nuclear arms and that is its enduring value."

"It has brought our two peoples closer together. We are entering the new year with a hope for continued progress, progress toward a safer world," Gorbachev said.

The Gorbachev videotape showed the Soviet leader, like Reagan, in a dark suit and dark tie. He was seated at a desk in an elegant setting, with light blue draperies in the background.